Sliding an edge while keeping connected 90 faces in same orientation?
Those are two different cases... In the first screenshot it's impossible to slide edge without affecting surrounding faces because of triangulated mesh. And as for the second one it's also impossible as you can only slide one edge (well, effectively).

If you haven't applied Rotation to the object you can simply change Transform Orientation to Local and move along selected axis:



But if you have it applied there are two methods to do this - regarding second case (quad based):

Snap selection to Edge Loop.


Create additional Edge Loop and place it where you want to 'slide' your selection.
Turn on Snapping: Vertex.
Move selection to created Edge Loop.
Remove Doubles and Dissolve remaining unnecessary edge.




Mesh Align Plus add-on.

Grab add-on from here: Mesh Align Plus and install it.


Go to Mesh Align Plus tab in Tool Shelf and select Quick Directional Slide section. 
Uncheck Auto Grab Source from selected Vertices.
Grab Normal Coordinates.
Enter Multiplier value - it's small in my case.
Check Enable Experimental Mesh Ops.
Move selection by clicking Mesh Piece.